( \ Proceedings: Third International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics
ls.!l April 2-7, 1995; Volume Ill, St. Louis, Missouri
A Unified Procedure for Earth Pressure Calculations
Paper No. 4.03
Ashok K. Chugh
Bureau of Reclamation
Denver, Colorado, USA
SYNOPSIS: A unified procedure for calculating active and passive earth pressures on retaining structures for static and
seismic (pseudostatic) loading conditions is presented. The procedure is based on the limit equilibrium method, uses the
method of slices, and satisfies complete statics. Necessary equations for a typical slice are presented and a solution scheme
for solving them is discussed. A sample problem is included to indicate convenience of use of the proposed procedure and
accuracy of results obtained. The results are in terms of magnitude, direction, and location of lateral thrust on the wall;
however, distribution of earth pressure along the wall height is not obtained and the direction of lateral thrust is user
specified.
INTRODUCTION
In geotechnical engineering practice, the commonly used
procedure for estimating earth pressure on retaining
structures is to use the formula p = y h K, where y is the
unit weight of backfill material - its value properly adjusted
for the design seismic coefficient in the vertical direction, h
is the height of the retaining structure, and K is the earth
pressure coefficient. Symbolically, K = K0, KA, or Kp for
static conditions of at-rest, active, and passive earth
pressures respectively; and K = KAE or KPE for earthquake
conditions of active and passive earth pressures respectively.
Appropriate values for K are calculated from published
formulae or selected from tables of values or their plotted
graphs. Lateral thrust on a retaining structure is given by
the area of the triangle representing the linear pressure
distribution implied by the above relation.
Published
resources for obtaining values of earth pressure coefficients
incorporate several assumptions such as: the backfill is
homogeneous, isotropic, dry, cohesionless. and sloping at a
constant angle; the shear sUJt.ace is a single plane; and the
soil mass is on the verge of failure. that is. factor of
safety = I. These assumptions may not necessarily be met
by design problems in the field. Also, in the conventional
procedures, location of the lateral thmst on the retaining
structure is estimated: for static ea1th pressures it is taken
to be at 0.33 h from the base; and for dynamic earth
pressures, it is taken to lie between 0.4 h to 0.75 h
depending on the manner and extent of the wall movement.
Since the physical response of a wall is not known a priori.
engineering judgement and past experience are used to
select the location of the dynamic thmst. See references by
Teng (1962), Seed and Whitman (1970). and Ebeling and
Morrison (1992); references included in this paper are
representative but not a complete list of works on the
subject.
Since the above procedures for static and dynamic earth
pressure calculations are based on limit equilibrium theory,
it seems reasonable to develop a complete set of equations
and their solution shall yield corresponding earth pressure
results in te1ms of magnitude, direction, and location of the
thrust. These earth pressure equations are similar to the
ones for slope stability equations. Geotechnical engineers
routinely analyze stability of slopes with complex geometry,
material distributions. material strengths, pore pressure
conditions, and shear surfaces of circular, non-circular, or
mixed shapes with tension cracks, etc. Most of the slope
stability procedures are based on the method of slices and
are computerized. Thus, it is advantageous to solve earth
pressure problems for the field conditions using a slope
stability analysis computer program.
However, this
procedure cannot give distribution of earth pressure along
the height of retaining structure and the direction of earth
pressure must be specified by the user engineer.
The objective of this paper is to present an adaptation and
use of slope stability analysis procedure to estimate active
and pas~-;ive earth pressures on retaining structures for static
and seismic (pseudostatic) conditions.
The necessary
equations are presented to explain the earth pressure
calculations. A sample problem is included to illustrate the
application of the ideas presented and to demonstrate the
accuracy of results obtained. In classical terminology, the
proposed procedure may be viewed as an extension of the
trial wedge method via the method of slices.
It is important to mention that there have been several
advanced theoretical and experimental studies made to
determine the effects of soi !-structure interaction on the
performance of retaining structures during earthquake
1179
conditions. While the end results of these studies remain
inconclusive, efforts are always directed at providing the
design engineer with a desirable selection of seismic
coefficient value for use in the pseudostatic (MononobeOkabe) equations and a location for the seismic earth
pressure force on the retaining structure. While these are
important studies, the material presented in this paper does
not deal with any of the soil-structure interaction issues.
Casagrande (1973) pointed out that when retaining
structures perform satisfactorily despite erroneous earthpressure assumptions, it is primarily because a cautious and
adequate safety factor has been allowed for in the design.
Whitman (1990) remarked that the dynamic behavior of
gravity retaining walls is much more complicated than
envisioned in the simple physical and mathematical model
that leads to the Mononobe-Okabe equation. However, this
venerable equation, when used with proper choice of input
parameters and suitable safety factors, still provides a sound
basis for design of many retaining structures.
ACTIVE EARTH PRESSURE EQUATIONS
Figure 1 is a general description of an earth pressure
problem. For a typical vertical slice, abed, the forces acting
on it are shown in figure 1(b) for active earth pressure
condition. Fe = '). .W is a force, which corresponds to a
constant acceleration ')... times that of gravity, acting at an
inclination y to the horizontal and through the center of
mass of the slice; HL and HR are the hydrostatic forces
exerted by the subsurface water on the vertical boundaries
of the slice; and other forces acting on the slice are self
explanatory. Considering the static equilibrium of forces
shown on figure 1(b) and combining them with the MohrCoulomb strength equation leads to:
ZL
= zR
-
1
-----=,....-----------,:; *
cos (o-a{1 -
_!._ c'b seca - W sina
[F
+
P cos(a -
~){~
+
(HL - HR)
cosa {1
h1
= -ZR
ZL
b (tan o
h2 - -
(W cosa - U)tan <!>'
F
+
~
~
~)}
tan a tan<!>'}
tan(a - y)tan <!>'}]
- tan a
2
p cos ~
ZL coso
(1)
) [ZR
-
+
]
1
ZL
(h 1 tan ~ - e)
(H L h4 - H R hs)
ZL coso
Surface loa
p
.2_
+
Similarly, considering the moment equilibrium of forces
leads to the following:
Fe
COS"(
ZL coso
(a]
tan(o-a)tan <!>']
tan <!>' - tan(a -
-Fe cos(a - y) {1 +
+
Potential slide surface
of same geometric
configuration
~
(he + e1 tany)
(2)
Equations ( 1) and (2) are for active earth pressure under
seismic (pseudostatic) loading. Corresponding expressions
for passive earth pressure are given in equations (3) and (4)
in the Appendix. Also, passive earth pressure expressions
can be achieved by specifying F = - F; o = (21t - o); and y
= ±(1t - y) for downward and upward inertial force,
respectively, in equations (1) and (2). For static conditions,
set Fe= 0.
SOLUTION PROCEDURE
(b)
Figure 1-(a) General retaining wall problem description
(b) Forces acting on a typical slice for
pseudostatic analysis (active condition)
Equations (1) and (2) are in the form of recursive
relationships. The solution procedure is initiated by using
the known boundary conditions ZR and h 2 at the far end of
the shear surface, and for the assigned values of factor of
safety F and interslice force inclination o. Equations (1)
and (2) are used to calculate the ZL and h 1 for the first
slice. ZR and h2 for the second slice equal ZL and h 1 for
the first slice. This process is continued until the ~ and h 1
for the last slice are calculated. ZL and h 1 for slice j ending
at the heel of a retaining wall give, respectively, the
1180
magnitude and location of the total thrust per foot length of
the retaining structure. Inclination of this force is at the
user specified value of 0. The variation of the earth
pressure along the height of the wall is not determined by
this procedure. The above procedure has been implemented
in the computer program SSTAB2 (Chugh, 1992).
SAMPLE PROBLEM
The problem shown in figure 2 was studied for active and
passive earth pressures for static and pseudostatic loading
conditions using a modified version of computer program
SSTAB2. Table 1 lists the various cases analyzed. Table
2 lists the results. These results compare favorably with the
results given in Ebeling and Morrison (1993). Under the
GENERAL COMMENTS
1. Equations (1) through (4) are for backfills retained on
the right side of the retaining structure shown in
figure 1. Similar equations can be derived for
backfills retained on the left side.
2. Equations (1) through (4) are for estimating earth
pressure forces for a prescribed pore water pressure
condition.
Figure 2 - Sample Problem
3. Selection of seismic coefficients in the horizontal and
vertical directions should be made in consultation
with experienced engineers and seismologists.
Table 1. - Sample problem data.
Analysis Backfill description Surcharge Pressure loading
No.
load
conditions
q
Active/
Static/
so
~0 c'
$'0
u Ibs/sq ft Passive Dynamic
I
0 0 30 0 0
0
Active Static
2
0 0 30 0 0
0
Passive Static
3
6 0 30 3 0
0
Active Static
4
6 0 30 -3 0
0
Passive Static
5
6 0 30 3 0
0
Active Dynamic
6
6 0 30 3 0
0
Active Dynamic
7
0
6 0 30 -3 0
Passive Dynamic
8
0 0 35 17.5 0
0
Active Dynamic
0 0 35 17.5 •
0
9
Active Dynamic
0
10
0 0 35 17.5 ••
Active Dynamic
II
0 0 35 17.5 • 500 Active Dynamic
0
Passive Dynamic
12*** 0 0 35 0 0
4. Strength values assigned for the backfill soils must
be consistent with the displacement of the wall and
the backfill soil.
Large displacements or
accumulation of small displacements may lead to the
use of residual shear strengths of the backfill.
5. Use of a factor of safety in earth pressure
calculations for static and dynamic loading conditions
may include considerations, such as: (a) earth
pressure calculations for dynamic loading make sense
only if the retaining wall has a reserve of strength
after the static needs have been met, that is, retaining
structure is designed for static earth pressure using
F > I; (b) appropriate value of F can be used to
reflect: (1) the level of uncertainty in shear strength
data for the backfill, and (2) the concerns for
nondevelopment of fully active or fully passive earth
pressure condition in the backfill.
Seismic coeff.
data
i..
0
0
0
0
0
0
0
0
0.12 -33.82
0.12 33.82
0.12 33.82
0.20
0
0.20
0
0.20
0
0
0.1
0.32 21.80
• Hydrostatic full depth.
•• Hydrostatic 12-foot depth.
••• Batter angle i = 5°
Table 2. - Sample problem results.
Converted results
Computed results
Total
thrust
Analysis
Equivalent Fraction
No. (lbs per foot Location a
pressure
of wall
hl (ft)
hl/H
coeff.
length)
600
8,000
0.337
6.74
0.333
72,000
0.347
2
6.934 29.7°
3.0
8,312
6.970 56.9° 0.346
0.349
3
4
96,480
0.447
8.936 32.3° 4.020
9,554
8.420 52°
0.421
5
0.427
0.414
6
10,640
8.285 52°
0.415
7
90,140
9.01 30.7° 4.026
0.451
0.401
8
9,113
8.020 49.6° 0.380
9
20,560
5.819 29.1° 0.857
0.291
10
12,650
6.699 45°
0.527
0.335
11
15,440
5.25
0.643
0.263
59°
12
72,210
5.51
3.419
0.276
23°
6. For active or at-rest earth pressure, the engineer
should investigate shear surfaces which give
maximum lateral thrust values. However, for passive
earth pressure, the emphasis should be to look for a
shear surface which gives minimum lateral thrust.
7. The procedure presented gives results for lateral
thrust on a retaining structure for the specified
geometry, backfill materials, and loading conditions.
For static earth pressures, specify Fe = 0.
8. Regular slope stability analysis should be performed
to evaluate the stability for a potential shear surface
passing through the foundation materials under the
retaining structure.
1181
y
present practice, lateral pressure calculations are made
individually for the various loadings of interest, each
analysis giving the magnitude and location of the lateral
thrust for the corresponding condition. However, in the
procedure presented, the problem is considered as one whole
and the analysis yields the magnitude, direction, and
location of lateral thrust on the wall.
SUMMARY: The proposed procedure provides an effective
and efficient means for using the limit equilibrium method
to determine the magnitude, direction, and location of lateral
thrust on a retaining structure for static and seismic
(pseudostatic) conditions without resorting to charts and in
a manner analogous to slope stability analysis procedure.
The geometry of the retaining wall and backfill materials,
pore water pressure conditions, surface loads, geometry of
shear surface, soil strengths, reinforcement, seismic
coefficients in horizontal and vertical directions, and desired
factor of safety and angle of inclination of lateral thrust on
the wall can all be specified by the engineer to reflect the
field conditions and design needs. However, the limit
equilibrium method has its limits of applicability, and the
decision to use this method for determining earth pressures
for a particular job should be made by the engineer in
charge of the project.
APPENDIX: PASSIVE EARTH PRESSURE EQUATIONS
Figure 3 - Forces acting on a typical slice for
pseudostatic analysis (passive condition)
Figure 3 is the free body diagram of a typical slice for
passive earth pressure condition. Considering the static
equilibrium of forces and combining them with the MohrCoulomb strength equation leads to:
~=~+
cos (o
REFERENCES
c' b seca
[_.!..
F
Casagrande, L. (1973), Comments on Conventional Design
of Retaining Structures, Journal of the Soil Mechanics and
Foundations Division, ASCE, Vol. 99, No. SM2, pp. 181198.
+
Chugh, A. K. (1992), User Information Manual for Slope
Stability Analysis Program 'SSTAB2', U.S. Bureau of
Reclamation, Denver, Colorado.
Ebeling, R~ M. and E. M. Morrison (1992), The Seismic
Design of Waterfront Retaining Structures, U.S. Army
Corps of Engineers, Waterways Experiment Station,
Vicksburg, Mississippi.
+
+
~) {~
P cos (a -
~
a) [1 -
W sina
+
_.!..
F
tan (o
cosa {1 -
- Fe cos (a
+
~
~
+
a) tan <))']
(W co sa - U) tan <))'
tan <))' - tan
- (H L - HR)
y){l -
*
(~
- a)}
tan a tan <)>'}
tan (a
+
y) tan<)>'}]
(3)
Similarly, the moment equilibrium of the forces acting on
the slice, figure 3, gives:
Seed, H. B. and R. V. Whitman (1970), Design of Earth
Retaining Structures for Dynamic Loads, ASCE Specialty
Conference on Lateral Stresses in the Ground and Design of
Earth Retaining Structures, pp. 103-147.
- p cos ~ (h tan ~ - e)
3
ZL coso
1
- - - (HL h4 - HR hs)
ZL coso
Teng, W. C. (1962), Foundation Design, Prentice Hall, Inc.,
Englewood Cliffs, New Jersey.
Whitman, R. V., (1990) Seismic Design and Behavior of
Gravity Retaining Walls, ASCE Geotechnical Special
Publication No. 25 on Design and Performance of Earth
Retaining Structures, pp. 817-842.
(4)
1182